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### fourier_cn

fourier_cn takes four or five arguments : an expression expr depending of a variable, the name of this variable (for example x), the period T, an integer n and a real a (by default a = 0).
fourier_cn(expr,x,T,n,a) returns the Fourier coefficient cn of a functionf of variable x defined on [a, a + T[ by f (x) = expr and periodic of period T:

cn = f (x)e-2inxTdx

To simplify the computations, one should input assume(n,integer) before calling fourier_cn to specify that n is an integer.
Examples
• Find the Fourier coefficients cn of the periodic function f of period 2 and defined on [- 1;1[ by f (x) = x2.
Input, to have c0 :
fourier_cn(x^2,x,2,0,-1)
Output:
1/3
Input, to have cn :
assume(n,integer)
fourier_cn(x^2,x,2,n,-1)
Output:
2*(-1)^n/(pi^2*n^2)

• Find the Fourier coefficients cn of the periodic function f, of period 2, and defined on [0;2[ by f (x) = x2.
Input, to have c0 :
fourier_cn(x^2,x,2,0)
Output:
4/3
Input, to have cn :
assume(n,integer)
fourier_cn(x^2,x,2,n)
Output:
((2*i)*pi*n+2)/(pi^2*n^2)

• Find the Fourier coefficients cn of the periodic function f of period 2. and defined on [0;2.[ by f (x) = x2.
Input :
assume(n,integer)
fourier_cn(x^2,x,2*pi,n)
Output :
((2*i)*pi*n+2)/n^2
If you don't specify assume(n,integer), the output will not be simplified :
((2*i)*pi^2*n^2*exp((-i)*n*2*pi)+2*pi*n*exp((-i)*n*2*pi)+
(-i)*exp((-i)*n*2*pi)+i)/(pi*n^3)
You might simplify this expression by replacing exp((-i)*n*2*pi) by 1, input :
subst(ans(),exp((-i)*n*2*pi)=1)
Output :
((2*i)*pi^2*n^2+2*pi*n+-i+i)/pi/n^3
This expression is then simplified with normal, the final output is :
((2*i)*pi*n+2)/n^2
Hence for n 0, cn = . As shown in this example, it is better to input assume(n,integer) before calling fourier_cn.
We must also compute cn for n = 0, input :
fourier_cn(x^2,x,2*pi,0)
Output :
4*pi^2/3
Hence for n = 0, c0 = .
Remarks :
• Input purge(n) to remove the hypothesis done on n.
• Input about(n) or assume(n), to know the hypothesis done on the variable n.

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giac documentation written by Renée De Graeve and Bernard Parisse